Flat foldability of general crease patterns was first claimed to be hard for
over twenty years. In this paper we prove that deciding flat foldability
remains NP-complete even for box pleating, where creases form a subset of a
square grid with diagonals. In addition, we provide new terminology to
implicitly represent the global layer order of a flat folding, and present a
new planar reduction framework for grid-aligned gadgets.